Who was the first to consider that categories were semi-simplicial sets (and in particular groupoids were simplicial sets)?

I think there was a concept of nerve of a covering in algebraic topology before (maybe Alexandroff).

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    $\begingroup$ This would be better on hsm.stackexchange.com. $\endgroup$
    – LSpice
    Dec 5, 2021 at 2:58
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    $\begingroup$ @BenjaminSteinberg Segal and Grothendieck did not introduce nerves: they introduced the idea of characterizing which simplicial sets are isomorphic to nerves. It is definitely Čech who introduced nerves of coverings. $\endgroup$ Dec 5, 2021 at 15:22
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    $\begingroup$ @D.-C.Cisinski the question is asking who introduced nerves of categories not nerves of coverings which undoubtedly goes back to cech. $\endgroup$ Dec 5, 2021 at 15:24
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    $\begingroup$ @D.-C.Cisinski: Nerves of covers were introduced by Paul Alexandroff in his 1928 paper Über den allgemeinen Dimensionsbegriff und seine Beziehungen zur elementaren geometrischen Anschauung. $\endgroup$ Dec 5, 2021 at 20:24
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    $\begingroup$ DOI link for Alexandroff's article: doi.org/10.1007/BF01451612, (and free pdf link: maths.ed.ac.uk/~v1ranick/papers/alexdim.pdf). I wonder if he talked with Noether about this stuff... $\endgroup$
    – David Roberts
    Dec 6, 2021 at 1:25

1 Answer 1


In Peter Johnstone's 1977 "Topos theory" (p.48) the simplicial description of categories is attributed to Grothendieck and he cites the "Technique de la descente"-series of Bourbaki seminars 1959-62 for it. I guess what he has in mind is in particular prop.4.1 on page 108 of the third installment Préschémas quotients from 1961.

  • $\begingroup$ This seems to be supported by Grothendieck himself. In a note of "Pursuing stacks" section 16bis pg 52, he says $\endgroup$ Dec 6, 2021 at 3:48
  • $\begingroup$ The definition of the nerve functor, realizing a full embedding of (Cat) into \Simplexˆ, is given for the first time (I believe) in a Bourbaki talk of mine, which at that time wasn’t concerned at all with the topological interpretation of small categories, but with some rather formal prerequisites for the operation of “passage to quotient” of an object by a “pre-equivalence relation”, in any category (with a view of applying it in the category of schemes. . . ) $\endgroup$ Dec 6, 2021 at 3:48

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